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1 Chapte 1 Intoduction This set of notes summaises the main esults of the lectue Elasticity (MATH35021). Please any coections (yes, thee might be the odd typo...) o suggestions fo impovement to Andew.Hazel@mancheste.ac.uk o see me afte the lectue. Geneally, the notes will be handed out afte the mateial has been coveed in the lectue. You can also download them fom the WWW: ahazel/math35021/math35021.html. This WWW page will also contain announcements, example sheets, solutions, etc. 1.1 Liteatue The following is a list of books that I found useful in pepaing this lectue. I ve quoted the pices whee I knew what they wee. It is not necessay to puchase any of these books! You lectue notes and these handouts will be completely sufficient. Textbook which coves most of the mateial in this lectue: Gould, P.L. Intoduction to Linea Elasticity, 2nd ed. Spinge (1994) 51 Nice (useful) eview of Linea Algeba: Banchoff, T. & Weme, J. Linea Algeba Though Geomety, 2nd ed. Spinge (1991). One of the classic elasticity texts: Geen, A.E. & Zena, W. Theoetical Elasticity. Dove (1992) papeback epint of the oiginal vesion fom Oxfod Univesity pess And anothe classic: Love, A.E.H. Teatise on the Mathematical Theoy of Elasticity. Dove (1944) papeback epint of the oiginal vesion fom Cambidge Univesity pess A beautiful little book (but out of pint!): Long, R.R. Mechanics of Solids and Fluids. Pentice- Hall, (1961) (back then, pesumably...). Ty the libay. My own favouite elasticity book (this book saved my PhD!): Wempne, G. Mechanics of Solids with Applications to Thin Bodies. Kluwe Academic Publishes Goup (1982) unfotunately only available as hadback fo 126!! 1.2 Peliminaies: Index notation & summation convention Denote vectos/matices/tensos by thei components, i.e. = i ; A = A ij Geek indices ange fom 1 to 2; Latin ones fom 1 to 3. { 1 fo i = j Konecke Delta: δ ij = 0 fo i j Summation convention: Automatic summation ove epated indices. E.g.: 1

2 MT30271 Elasticity: Intoduction 2 Dot poduct: a b = a i b i = a k b k (Dummy index!) δ ij exchanges indices: a i δ ij = a j. Matix-vecto poducts: A x = A ij x j = A im x m A T x = A ji x j No summation ove indices in backets: E.g. diagonal matix: diag(λ 1, λ 2, λ 3 ) = λ (i) δ (i)j. Comma denotes patial diffeentiation: E.g. Some diffeential opeatos in index notation: u i x j = u i,j. u = div u = u i,i (1.1) φ = gadφ = φ,i (1.2) 2 φ = φ,ii (1.3)

3 Chapte 2 Analysis of stain 2.1 The infinitesimal stain tenso defomed configuation 00 undefomed efeence 00 configuation u+ d00 u Q q P p d R 00 d 00 u R Figue 2.1: Sketch illustating the defomation of an elastic body: The body is displaced, otated and defomed. Lagangian desciption: Label mateial points by thei coodinates befoe the defomation (i.e. in the efeence configuation). Displacement field: The mateial paticle at position i = x i befoe the defomation is displaced to R i afte the defomation: R i = i + u i (x j ). (2.1) The defomation changes mateial line elements fom d i (= dx i ) to dr i : dr i = d i + u i x j dx j. (2.2) We will estict ouselves to a lineaised anlysis in which the displacement deivatives ae small, i.e. u i x j 1. (2.3) ui x j is the displacement gadient tenso: whee u i x j = e ij + ω ij, (2.4) e ij = 1 ( ui + u ) j = e ji is the stain tenso and (2.5) 2 x j x i ω ij = 1 ( ui u ) j = ω ji is the otation tenso. (2.6) 2 x j x i 3

4 MT30271 Elasticity: Analysis of stain 4 Displacements in the vicinity of : u i ( + d) = u i () } {{ } Rigid Body Tanslation + ω ij dx j } {{ } Rigid Body Rotation + e ij dx j } {{ } Pue Defomation (2.7) 2.2 Rigid body otation Fo e ij = 0 (2.2) and (2.7): dr = d + ω d (2.8) whee ω = (ω 32, ω 13, ω 21 ) T. Repesents igid body otation fo ω ij Pue defomation Extensional defomation Duing the defomation the line element d i = ds n i is stetched to dr i = ds N i (n and N ae unit vectos). undefomed efeence configuation q p d = ds n defomed configuation Q P dr= ds N Figue 2.2: Sketch illustating the extension (and otation) of mateial line elements duing the defomation of an elastic body. The nomal stain e n is the elative extension of the line element ds n: e n = ds ds ds = e ij n i n j (2.9) The e (i)(i) ae the nomal stains along the coodinate axes Shea defomation Conside the change of the angle between two mateial line elements d (1) = ds (1) n (1), d (2) = ds (2) n (2) which ae othogonal to each othe in the undefomed state, (d (1) i d (2) i = 0). Befoe the defomation: ϕ = π/2. Afte the defomation (see Fig. 2.3): cosφ = 2e ij n (1) i n (2) j. (2.10) The e ij fo i j ae the shea stains w..t. the coodinate axes. 2.4 Pincipal axes/stain invaiants The stain tenso gives the stains elative to the chosen coodinate system. Rotation of the coodinate system to a new one, such that x i = a ij x j whee a ij a kj = δ ik (othogonal matix, A T = A 1 ) (2.11)

5 MT30271 Elasticity: Analysis of stain 5 undefomed efeence configuation ϕ = π/2 0 1 (1) 0 1 d d (2) dr defomed configuation (1) φ dr (2) Figue 2.3: Sketch illustating the shea defomation, i.e. the change in the angle between two mateial line elements duing the defomation of an elastic body. tansfoms the components of the stain tenso to: e ij = a ik e kl a jl (symbolically Ẽ = AEA T ). (2.12) Thee exists a special coodinate system (pincipal axes) in which e ij = 0 fo i j. The pincipal axes ae the nomalised eigenvectos of e ij. The nomalised eigenvectos fom the ows of the tansfomation matix a ik to the coodinate system fomed by the pincipal axes. The eigenvalues of e ij ae the pincipal stains, i.e. the stains in the diections of the nomal axes. The maximum nomal stain, maxe n, (max. ove all diections n) is given by the maximum pincipal stain. The stain tenso has thee invaiants (i.e. quantities that ae independent of the choice of the coodinate system): the dilation: d = e ii which epesents the elative change in volume the deteminant: dete ij. and a thid quantity: 1/2(e ij e ij e ii e jj ) 2.5 Stain compatibility d = e ii = (dv dv)/dv (2.13) Equation (2.5) expesses e ij in tems of a given displacement field u i. The invese poblem: e ij only descibes a continuous defomation of a body (i.e. no gaps o ovelaps of mateial develop duing the defomation) iff: e ij,kl + e kl,ij e kj,il e il,kj = 0 (2.14) This epesents 3 4 = 81 equations but only the ones coesponding to the following six paamete combinations ae non-tivial and distinct: i j k l Geometical intepetation which motivates the deivation of eqns. (2.14): e ij detemines the defomation of infinitesimal ectangula (cubic in 3D) blocks of mateial. Afte the defomation, the individually defomed blocks of mateial (defomed accoding to thei local value of e ij ) must still fit togethe to fom a continuous body.

6 MT30271 Elasticity: Analysis of stain 6 undefomed efeence configuation path (i) b path (I) defomed configuation B a A path (ii) path (II) Figue 2.4: Sketch illustating the stain compatibility condition. 2.6 Homogeneous defomation A defomation fo which thoughout the body is called a homogeneous defomation. Examples: Simple extension E.g. e 11 = e 0, e ij = 0 othewise. Unifom dilation e ij = e 0 δ ij (spheically symmetic). Simple sheaing E.g. e 12 = e 21 = e 0, e ij = 0 othewise. u i x j = const. (2.15)

7 Chapte 3 Analysis of stess 3.1 The concept of taction/stess If F is the esultant foce acting on a small aea element S with unit nomal n, then the taction (stess) vecto t is defined as: F t = lim (3.1) S 0 S The tem taction is usually used fo stesses acting on the sufaces of a body. F n S S F n Figue 3.1: Sketch illustating taction and stess. 3.2 The stess tenso The stess vecto t depends on the spatial position in the body and on the oientation of the plane (chaacteised by the nomal vecto): t i = τ ij n j, (3.2) whee τ ij = τ ji is the stess tenso. On an infinitesimal block of mateial whose faces ae paallel to the axes, the component τ ij of the stess tenso epesents the taction component in the positive i-diection on the face x j = const. whose nomal points in the positive j-diection (see Fig. 3.2). 3.3 The equations of equilibium/motion The equations of equilibium fo a body, subject to a body foce (foce pe unit volume) F i is τ ij x j + F i = 0. (3.3) Including inetial effects via D Alembet foces gives the equations of motion: τ ij x j + F i = ρ 2 u i t 2, (3.4) 7

8 MT30271 Elasticity: Analysis of stess 8 x 3 x 3 x τ τ τ τ τ τ τ τ12 τ x 1 2 x τ τ τ τ τ 32 τ τ τ τ x 1 2 Figue 3.2: Sketch illustating the components of the stess tenso. whee ρ is the density of the body and t is time. 3.4 Pincipal axes/stess invaiants The stess tenso is eal and symmetic, hence all consideations in section 2.4 apply to the stess tenso as well (tansfomation to diffeent coodinate systems, pincipal axes, max. stess and invaiants). In paticula, we will denote the fist invaiant (the tace of the stess vecto) by 3.5 Homogeneous stess states Analogous to homogeneous defomations (see section 2.6): Examples: Uniaxial stess E.g. τ 11 = T 0, τ ij = 0 othewise. Hydostatic pessue τ ij = P 0 δ ij (spheically symmetic). Pue shea stess E.g. τ 12 = τ 21 = T 0, τ ij = 0 othewise. θ = τ ii. (3.5)

9 Chapte 4 Elasticity & constitutive equations 4.1 The constitutive equations The constitutive equations detemine the stess τ ij in the body as function of the body s defomation. Definition: A solid body is called elastic if τ ij (x n, t) = τ ij (e kl (x n, t)). (4.1) i.e. if the stess depends on the instantaneous, local values of the stain only. Fo small stains, a Taylo expansion of (4.1) gives: τ ij = τ ij ekl =0 + τ ij } {{ } e kl e kl. (4.2) ekl =0 Initial Stess τij 0 } {{ } E ijkl If the efeence configuation coincides with a stess fee state, then τij 0 = 0 and we obtain Hooke s law: τ ij = E ijkl e kl. (4.3) Definition: A solid body is called homogeneous if E ijkl is independent of x i. Definition: A solid body is called isotopic if its elastic popeties ae the same in all diections. Fo an isotopic homogeneous elastic solid: whee λ and µ ae the Lamé constants. E ijkl = λδ ij δ kl + 2µδ ik δ jl, (4.4) Stess-stain elationship fo an isotopic homogeneous elastic solid: and in the invese fom: so that Witten out: τ ij = λδ ij e }{{} kk +2µe ij, (4.5) =d e ij = 1 ( ) λ δ ik δ jl 2µ (3λ + 2µ) δ ijδ kl τ kl (4.6) } {{ } D ijkl e ij = D ijkl τ kl. (4.7) e ij = 1 2µ τ λ ij 2µ(3λ + 2µ) δ ij τ }{{} kk =θ (4.8) 9

10 MT30271 Elasticity: Elasticity & constitutive equations 10 Fo an isotopic homogeneous elastic solid the pincipal axes of the stess and stain tensos coincide and θ = τ kk = (3λ + 2µ)d = (3λ + 2µ)e kk (4.9) 4.2 Expeimental detemination of elastic constants I. Simple Extension L D T D+ D L+ L x 2 II. Simple Shea γ τ 00 x 3 00 T x 1 Figue 4.1: Sketch illustating the two fundamental expeiments fo the detemination of the elastic constants Expeiment I: Simple extension of a thin cylinde Obsevations: i.e. (since e 33 = L/L) and whee e 11 = e 22 = D/D. T = EA L L E and ν ae Young s modulus and Poisson s atio, espectively Expeiment II: Simple shea Obsevation: i.e. G is the mateial s shea modulus. (4.10) τ 33 = Ee 33 (4.11) e 11 e 33 = e 22 e 33 = ν (4.12) τ = Gγ (4.13) τ 12 = G 2e 12. (4.14) Constitutive equations in tems of E and ν τ ij = E e ij + ν 1 + ν 1 2ν δ ij e }{{} kk. (4.15) Note that mateials with ν = 1/2 ae incompessible, i.e. d 0. d e ij = 1 (1 + ν)τ ij νδ ij τ kk. (4.16) E }{{} θ

11 MT30271 Elasticity: Elasticity & constitutive equations Relations between the elastic constants λ = µ = G = E = ν = λ, µ λ µ λ(1 2ν) 2ν µ(3λ+2µ) λ+µ (1+ν)(1 2ν)λ ν λ 2(λ+µ) λ, ν λ ν µ(e 2µ) µ, E 3µ E µ E Eν E E, ν (1+ν)(1 2ν) 2(1+ν) E ν E 2µ 2µ

12 Chapte 5 The equations of linea elasticity 5.1 Summay of equations Stain-displacement elations: e ij = 1 2 (u i,j + u j,i ) (5.1) Equilibium equations/equations of motion: Constitutive equations: τ ij,j + F i = ρ 2 u i t 2 (5.2) τ ij = λδ ij e kk + 2µe ij (5.3) 5.2 Displacement fomulation: The Navie-Lamé equations Solve fo the displacements: o symbolically: which is equivalent to: (λ + µ)u k,ki + µu i,kk + F i = ρ 2 u i t 2 (5.4) (λ + µ) gad div u + µ 2 u + F = ρ 2 u t 2, (5.5) (λ + 2µ) gad div u µ cul cul u + F = ρ 2 u t 2. (5.6) This is a system of thee coupled linea elliptic PDEs fo the thee displacements u i (x j ). 5.3 Stess fomulation: The static Beltami-Mitchell equations Fo static defomations, we have 1 ν 1 + ν τ ii,jj +F i,i = 0 o symbolically }{{} θ,jj 1 ν 1 + ν 2 θ + div F = 0. (5.7) and the stesses fulfil the Beltami-Mitchell equations: 1 τ ij,kk + } {{ } 1 + ν τ ν kk,ij + } {{ } 1 ν δ ij F k,k +F j,i + F i,j = 0. (5.8) }{{} 2 τ ij θ,ij div F (5.8) epesents a system of six coupled linea elliptic PDEs fo the six stess components τ ij (x j ). When these have been detemined, the stains can be ecoveed fom (4.6) o (4.16). Then the displacements follow fom (5.1). They ae only detemined up to abitay igid body motions. 12

13 MT30271 Elasticity: The equations of linea elasticity Simplifications fo F = const.: Fo constant (o vanishing!) body foce, the stess, stain and displacement components ae bihamonic functions, u i,jjkk = 0 τ ij,kkll = 0 e ij,kkll = 0 (5.9) o symbolically: 4 u = 0 4 τ ij = 0 4 e ij = 0. (5.10) The dilation and the tace of the stess tenso ae hamonic functions: u j,jkk = d,kk = 0 τ jj,kk = θ,kk = 0 (5.11) o symbolically: 2 d = 0 2 θ = 0 (5.12) Note that in (5.4) (5.8) F acts as an inhomogeneity in a system of linea equations. The system can be tansfomed into a homogeneous system fo u h = u u p (with diffeent bounday conditions) if a paticula solution u p (which does not have to fulfil the bounday conditions) can be found. 5.5 Bounday conditions: Displacement (Diichlet) bounday conditions: Pescibed displacement field u (0) i. u i D = u (0) i (5.13) Stess (Neumann) bounday conditions: Pescibed (applied) taction t (0) i on bounday. Note that n j is the oute unit nomal vecto on the elastic body. t i D = τ ij n j D = t (0) i (5.14) Mixed (Robin) bounday conditions elastic foundation epesented by the stiffness tenso k ij. Physically, this implies that the taction which the elastic foundation exets on the body is popotional to the bounday displacement. This can be combined with an applied taction t (0) i as in the Neumann case. (t i + k ij u j ) D = (τ ij n j + k ij u j ) D = t (0) i (5.15)

14 MT30271 Elasticity: The equations of linea elasticity 14 Govening Equations in Cylindical Pola Coodinates x 1 = x = cosθ, x 2 = y = sin θ, x 3 = z = z. Vecto calculus: u = (u, u θ, u z ), e = (e ij ), τ = (τ ij ), whee i, j =, θ, z. gadf = f ˆ + 1 culu = ( 1 u z θ u θ z f θ ˆθ + f z ẑ, ) ˆ + ( u z u z divu = 1 (u ) + 1 ) ˆθ + ( 1 (u θ ) 1 Stess-stain elations have the same fom as in Catesian coodinates: τ ij = λδ ij divu + 2µe ij, i, j =, θ, z. u θ θ + u z z, u θ ) ẑ. Stess-displacement elations: τ = λdivu + 2µ u ( 1, τ u θ θθ = λdiv u + 2µ θ + u ), τ zz = λdivu + 2µ u z z, τ θ µ = τ θ µ = u θ u θ + 1 u θ, Stain-displacement elations: τ z µ = τ z µ = u z + u z, τ θz µ = τ zθ µ = 1 u z θ + u θ z. e = u, e θθ = 1 u θ θ + u, e zz = u z z, 2e θ = 2e θ = u θ u θ + 1 u θ, 2e z = 2e z = u z + u z, 2e zθ = 2e θz = 1 Equilibium equations (statics): fo the displacement fomulation, use Navie s equation, wheeas fo the stess fomulation, use τ (λ + 2µ)gaddivu µ culculu + F = 0, + 1 τ θ τ θ θ + τ z z + τ τ θθ τ θθ + F = θ + τ θz z + 2 τ θ + F θ = 0 τ z + 1 τ θz θ + τ zz z + 1 τ z + F z = 0. Stess bounday conditions: these ae when t is pescibed. We have, fom t i = ˆn j τ ij, t = ˆn τ + ˆn θ τ θ + ˆn z τ z t θ = ˆn τ θ + ˆn θ τ θθ + ˆn z τ θz t z = ˆn τ z + ˆn θ τ θz + ˆn z τ zz u z θ + u θ z.

15 MT30271 Elasticity: The equations of linea elasticity 15 Govening Equations in Spheical Pola Coodinates x 1 = x = sinθ cosφ, x 2 = y = sin θ sinφ, x 3 = z = cosθ. u = (u, u θ, u φ ), e = (e ij ), τ = (τ ij ), whee i, j =, θ, φ. Vecto calculus: divu = gadf = f ˆ + 1 { 1 2 sinθ f θ ˆθ + 1 f sin θ φ ˆφ, (2 sin θ u ) + θ ( sin θ u θ) + 1 ˆ ˆθ sin θ ˆφ culu = 2 sinθ θ φ u u θ sin θ u. φ } φ (u φ), Stess-stain elations have the same fom as in Catesian coodinates: Stess-displacement elations: τ ij = λδ ij divu + 2µe ij, i, j =, θ, φ. τ = λdivu + 2µ u, τ θθ = λdivu + 2µ τ φφ = λdivu + 2µ ( ) 1 u φ sin θ φ + u + u θ cotθ, τ φ µ = τ φ µ = 1 u sinθ φ + u φ u φ, Stain-displacement elations: e = u, ( ) uθ θ + u, τ θ µ = τ θ µ = u θ u θ + 1 u θ, τ θφ µ = τ φθ µ = 1 u θ sinθ φ + 1 u φ θ u φ cotθ. e θθ = 1 u θ θ + u, e φφ = 1 u φ sin θ φ + u + u θ cotθ, 2e θ = 2e θ = u θ u θ + 1 u θ, 2e φ = 2e φ = 1 u sin θ φ + u φ u φ, 2e φθ = 2e θφ = 1 u θ sin θ φ + 1 u φ θ u φ cotθ. Equilibium equations (statics): fo the displacement fomulation, use Navie s equation, wheeas fo the stess fomulation, use τ + 1 τ θ + 1 (λ + 2µ)gaddivu µ culculu + F = 0, τ θ θ + 1 sin θ τ θθ θ + 1 sin θ τ φ + 1 τ φ φ + 2τ τ θθ τ φφ + cotθ τ θ τ θφ θ + 1 sin θ τ θφ φ + 3τ θ + (τ θθ τ φφ )cotθ + F = 0 + F θ = 0 τ φφ φ + 3τ φ + 2τ θφ cotθ + F φ = 0. Stess bounday conditions: these ae when t is pescibed. We have, fom t i = ˆn j τ ij, t = ˆn τ + ˆn θ τ θ + ˆn φ τ φ t θ = ˆn τ θ + ˆn θ τ θθ + ˆn φ τ θφ t φ = ˆn τ φ + ˆn θ τ θφ + ˆn φ τ φφ

16 Chapte 6 Plane stain poblems 6.1 Basic equations Definition: A defomation is said to be one of plane stain (paallel to the plane x 3 = 0) if: u 3 = 0 and u α = u α (x β ). (6.1) Thee ae only two independent vaiables, (x 1, x 2 ) = (x, y). Plane stain is only possible if F 3 = 0. Only the in-plane stains ae non-zeo, e i3 = 0. Stess-stain elationship: Static equilibium equations: τ αβ = λδ αβ e γγ + 2µe αβ. (6.2) 2µe αβ = τ αβ νδ αβ τ γγ }{{} θ (6.3) τ 33 = ντ γγ = ν θ (6.4) τ αβ,β + F α = 0 (6.5) Compatibility equation: Only one non-tivial equation 0 = e 11,22 + e 22,11 2e 12,12 (6.6) Fomulated in tems of stesses: o symbolically whee 2 = 2 / x / y 2. (1 ν) θ,αα + F α,α = 0, (6.7) (1 ν) 2 θ + div F = 0, (6.8) 6.2 The Aiy stess function Fo F = 0 the in-plane stesses can be expessed in tems of the Aiy stess function φ: τ 11 = 2 φ y 2, The Aiy stess function is bihamonic: τ 22 = 2 φ x 2, τ 12 = 2 φ x y. (6.9) 4 φ = 4 φ x φ x 2 y φ = 0. (6.10) y4 16

17 MT30271 Elasticity: Plane stain poblems The stess bounday conditions in tems of the Aiy stess function The applied tactions along the bounday D (paametised by the aclength s) ae given in tems of the Aiy stess function φ by t 1 (s) = t x (s) = d ( ) φ (6.11) ds y and t 2 (s) = t y (s) = d ds ( ) φ. (6.12) x Hence, if t α (s) is given, the bounday conditions fo φ can be deived by the following pocedue: 1. Integate (6.11) and (6.12) along the bounday (w..t. s). This povides ( φ/ x, φ/ y) T = φ on the bounday. 2. Rewite φ = φ/ s e t + φ/ n e n whee e t and e n ae the unit tangent and (oute) nomal vectos on the bounday. This povides φ/ s and φ/ n along the bounday. 3. Integate φ/ s along the bounday (w..t. s). This povides φ along the bounday. Afte this pocedue φ and φ/ n ae known along the entie bounday and can be used as the bounday condition fo the fouth ode bihamonic equation (6.10). Note: any constants of integation aising duing the pocedue can be set to zeo. Fo a taction fee bounday, t α (s) = 0, we can use the bounday conditions: φ = 0 and φ/ n = 0 on D (6.13) 6.4 The displacements in tems of the Aiy stess function Fo a given Aiy stess function φ(x, y), the displacements u(x, y), v(x, y), ae detemined by 2µ u x = (1 ν) 2 φ 2 φ x 2, (6.14) and 2µ v y = (1 ν) 2 φ 2 φ y 2 (6.15) ( u µ y + v ) = 2 φ x x y. (6.16) One way to detemine the displacement fields fom these equations is given by the following pocedue: 1. Get p(x, y) = 2 φ(x, y) fom the known φ(x, y). 2. p(x, y) is a hamonic function; detemine its complex conjugate q(x, y) fom the Cauchy- Riemann equations: p x = q p and y y = q x. (6.17) 3. Integate f(z) = f(x + iy) = p(x, y) + i q(x, y) and thus detemine P(x, y) and Q(x, y) fom F(z) = f(z)dz =: P(x, y) + i Q(x, y). (6.18)

18 MT30271 Elasticity: Plane stain poblems Then the displacements ae given by: u(x, y) = 1 [ (1 ν) P(x, y) φ 2µ x + and v(x, y) = 1 [ (1 ν) Q(x, y) φ 2µ y Equations in pola coodinates a + cy } {{ } igid body motion b cx } {{ } igid body motion ] (6.19) ]. (6.20) The bihamonic equation in pola coodinates: [ 4 2 φ(, ϕ) = ϕ 2 ] [ 2 φ φ ] φ 2 ϕ 2 (6.21) 4 φ(, ϕ) = φ, + 2 φ, 1 2 (φ, 2φ,ϕϕ ) (φ, 2φ,ϕϕ ) (4φ,ϕϕ + 2φ,ϕϕϕϕ ) (6.22) Fo axisymmetic solutions: 4 φ() = 1 [ ( ) ] 1 [φ,],,, (6.23) 4 φ() = φ, + 2 φ, 1 2 φ, φ, (6.24) Stesses: τ = 1 2 φ 2 ϕ φ, (6.25) τ ϕϕ = 2 φ 2 (6.26) τ ϕ = 1 2 φ ϕ 1 2 φ ϕ = ( 1 φ ϕ ). (6.27) 6.6 Paticula solutions of the bihamonic equation Hamonic functions Obviously, all hamonic functions also fulfil the bihamonic equation Powe seies expansions φ(x, y) = i,k a ik x i y k (6.28) Any tems with i + k < 2 do not give a contibution. Any tems with i + k < 4 fulfil 4 φ = 0 fo abitay constants a ik. Special cases ae: φ(x, y) τ xx τ yy τ xy Intepetation: a 02 y 2 2 a constant tension in x-diection a 11 xy 0 0 a 11 pue shea a 20 x a 20 0 constant tension in y-diection a 03 y 3 6 a 03 y 0 0 pue x-bending a 30 x a 30 x 0 pue y-bending Linea combinations povide stess fields fo combined load cases.

19 MT30271 Elasticity: Plane stain poblems Solutions in pola coodinates The geneal axisymmetic solution: φ() = A 0 + B C 0 ln + D 0 2 ln (6.29) The geneal sepaated non-axisymmetic solution: Fo n = 1: ( φ(, ϕ) = A + B ) + C3 + D log ( + a + b ) + c3 + d log cos(ϕ) sin(ϕ) (6.30) Fo n 2: φ(, ϕ) = ( An n + B n n + C n n+2 + D n n+2) cos(nϕ) n=2 6.7 St. Venant s pinciple + ( a n n + b n n + c n n+2 + d n n+2) sin(nϕ) (6.31) Section 6.6 povides many solutions of the bihamonic equation. The fee constants in these solutions have to be detemined fom the bounday conditions. This is the hadest pat of the solution! Good appoximate solutions can often be obtained by using: St. Venant s pinciple In elastostatics, if the bounday tactions t on a pat D 1 of the bounday D ae eplaced by a statically equivalent taction distibution ˆt, the effects on the stess distibution in the body ae negligible at points whose distance fom D 1 is lage compaed to the maximum distance between the points of D 1. Statically equivalent means that the esultant foces and moments due to the two tactions t and ˆt ae identical. Hence, the taction bounday conditions ae not fulfilled pointwise but in an aveage sense.